Find the value of `sin 75^(@) sin 15 ^(@).`

Answer

To find the value of \( \sin 75^\circ \sin 15^\circ \), we can follow these steps:

### Step 1: Calculate \( \sin 75^\circ \)
We can express \( 75^\circ \) as \( 45^\circ + 30^\circ \). Using the sine addition formula:
\[ \sin(a + b) = \sin a \cos b + \cos a \sin b \]
we have:
\[ \sin 75^\circ = \sin(45^\circ + 30^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ \]

### Step 2: Substitute known values
Using known values:
- \( \sin 45^\circ = \frac{1}{\sqrt{2}} \)
- \( \cos 30^\circ = \frac{\sqrt{3}}{2} \)
- \( \cos 45^\circ = \frac{1}{\sqrt{2}} \)
- \( \sin 30^\circ = \frac{1}{2} \)

Substituting these values, we get:
\[ \sin 75^\circ = \left(\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2}\right) + \left(\frac{1}{\sqrt{2}} \cdot \frac{1}{2}\right) \]
\[ = \frac{\sqrt{3}}{2\sqrt{2}} + \frac{1}{2\sqrt{2}} = \frac{\sqrt{3} + 1}{2\sqrt{2}} \]

### Step 3: Calculate \( \sin 15^\circ \)
We can express \( 15^\circ \) as \( 45^\circ - 30^\circ \). Using the sine subtraction formula:
\[ \sin(a - b) = \sin a \cos b - \cos a \sin b \]
we have:
\[ \sin 15^\circ = \sin(45^\circ - 30^\circ) = \sin 45^\circ \cos 30^\circ - \cos 45^\circ \sin 30^\circ \]

### Step 4: Substitute known values
Substituting the same known values:
\[ \sin 15^\circ = \left(\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2}\right) - \left(\frac{1}{\sqrt{2}} \cdot \frac{1}{2}\right) \]
\[ = \frac{\sqrt{3}}{2\sqrt{2}} - \frac{1}{2\sqrt{2}} = \frac{\sqrt{3} - 1}{2\sqrt{2}} \]

### Step 5: Multiply \( \sin 75^\circ \) and \( \sin 15^\circ \)
Now, we multiply the two results:
\[ \sin 75^\circ \sin 15^\circ = \left(\frac{\sqrt{3} + 1}{2\sqrt{2}}\right) \left(\frac{\sqrt{3} - 1}{2\sqrt{2}}\right) \]
\[ = \frac{(\sqrt{3} + 1)(\sqrt{3} - 1)}{4 \cdot 2} \]
Using the difference of squares:
\[ = \frac{(\sqrt{3})^2 - (1)^2}{8} = \frac{3 - 1}{8} = \frac{2}{8} = \frac{1}{4} \]

### Final Answer
Thus, the value of \( \sin 75^\circ \sin 15^\circ \) is \( \frac{1}{4} \).

---